In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric
immersion of a real-space-form Mn(c) of constant sectional curvature c into a
complex-space-form
M˜n(4c) of constant sectional curvature 4c associated with each twisted
product decomposition of a real-space-form if its twistor form is twisted closed.
Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric
immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c)
admits an appropriate twisted product decomposition with twisted closed twistor
form and, moreover, the immersion L is determined by the corresponding adapted
Lagrangian isometric immersion of the twisted product decomposition. It is natural
to ask the explicit expressions of adapted Lagrangian isometric immersions of
twisted product decompositions of real-space-forms Mn(c) into complex-space-forms
M˜n(4c) for each case: c = 0, c > 0 and c < 0.